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Theorem bj-gl4lem 32908
Description: Lemma for bj-gl4 32909. Note that this proof holds in the modal logic (K). (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-gl4lem (∀𝑥𝜑 → ∀𝑥(∀𝑥(∀𝑥𝜑𝜑) → (∀𝑥𝜑𝜑)))

Proof of Theorem bj-gl4lem
StepHypRef Expression
1 19.26 1947 . . 3 (∀𝑥(∀𝑥𝜑𝜑) ↔ (∀𝑥𝑥𝜑 ∧ ∀𝑥𝜑))
2 simpr 479 . . . . 5 ((∀𝑥𝑥𝜑 ∧ ∀𝑥𝜑) → ∀𝑥𝜑)
32a1i 11 . . . 4 (𝜑 → ((∀𝑥𝑥𝜑 ∧ ∀𝑥𝜑) → ∀𝑥𝜑))
43anc2ri 582 . . 3 (𝜑 → ((∀𝑥𝑥𝜑 ∧ ∀𝑥𝜑) → (∀𝑥𝜑𝜑)))
51, 4syl5bi 232 . 2 (𝜑 → (∀𝑥(∀𝑥𝜑𝜑) → (∀𝑥𝜑𝜑)))
65alimi 1888 1 (∀𝑥𝜑 → ∀𝑥(∀𝑥(∀𝑥𝜑𝜑) → (∀𝑥𝜑𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886
This theorem depends on definitions:  df-bi 197  df-an 385
This theorem is referenced by:  bj-gl4  32909
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